In this section we generalize the definition of Boreale's linear weighted
bisimulation [1]} from a field to a semiring \(K\) . We show
that the two notions coincide in the special case when \(K\)  is a field, for
example \(K =\mathbb {R}\) , as in [1]}. Starting with this section
almost all the results are our contribution and involve new concepts and proofs.
